Hypergraphs with no tight cycles

We show that every $r$-uniform hypergraph on $n$ vertices which does not contain a tight cycle has at most $O(n^{r-1} (\log n)^5)$ edges. This is an improvement on the previously best-known bound, of $n^{r-1} e^{O(\sqrt{\log n})}$, due to Sudakov and Tomon, and our proof builds up on their work. A recent construction of B. Janzer implies that our bound is tight up to an $O((\log n)^4 \log \log n)$ factor.
Comment: 9 pages; corrected typos